Quasi-exactly solvable quartic: elementary integrals and asymptotics

We study elementary eigenfunctions y=p exp(h) of operators L(y)=y"+Py, where p, h and P are polynomials in one variable. For the case when h is an odd cubic polynomial, we found an interesting identity which is used to describe the spectral locus. We also establish some asymptotic properties of the QES spectral locus.Comment: 20 pages, 1 figure. Added Introduction and several references, corrected misprint

Universality of Cluster Dynamics

We have studied the kinetics of cluster formation for dynamical systems of dimensions up to $n=8$ interacting through elastic collisions or coalescence. These systems could serve as possible models for gas kinetics, polymerization and self-assembly. In the case of elastic collisions, we found that the cluster size probability distribution undergoes a phase transition at a critical time which can be predicted from the average time between collisions. This enables forecasting of rare events based on limited statistical sampling of the collision dynamics over short time windows. The analysis was extended to L$^p$-normed spaces ($p=1,...,\infty$) to allow for some amount of interpenetration or volume exclusion. The results for the elastic collisions are consistent with previously published low-dimensional results in that a power law is observed for the empirical cluster size distribution at the critical time. We found that the same power law also exists for all dimensions $n=2,...,8$, 2D L$^p$ norms, and even for coalescing collisions in 2D. This broad universality in behavior may be indicative of a more fundamental process governing the growth of clusters

Approaching criticality via the zero dissipation limit in the abelian avalanche model

The discrete height abelian sandpile model was introduced by Bak, Tang & Wiesenfeld and Dhar as an example for the concept of self-organized criticality. When the model is modified to allow grains to disappear on each toppling, it is called bulk-dissipative. We provide a detailed study of a continuous height version of the abelian sandpile model, called the abelian avalanche model, which allows an arbitrarily small amount of dissipation to take place on every toppling. We prove that for non-zero dissipation, the infinite volume limit of the stationary measure of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure. We show that in the whole non-zero dissipation regime, the model is not critical, i.e., spatial covariances of local observables decay exponentially. We then study the zero dissipation limit and prove that the self-organized critical model is recovered, both for the stationary measure and for the dynamics. We obtain rigorous bounds on toppling probabilities and introduce an exponent describing their scaling at criticality. We rigorously establish the mean-field value of this exponent for $d > 4$.Comment: 46 pages, substantially revised 4th version, title has been changed. The main new material is Section 6 on toppling probabilities and the toppling probability exponen

Colliding cascades model for earthquake prediction

1.1 The model We explore here the process wherein seismicity undergoes a qualitative change, culminating in a major earthquake. This is done for synthetic seismicity generated by the model of collidin

Formal morphostructural zoning of mountain territories

Formalization of morphostructural zoning of mountain regions, with special emphasis on determination of lineaments, is considered in this paper. The zoning is based on joint analysis of topography, geology, tectonics and geomorphology, represented on corresponding maps and aerial or space photos. Our goal is to make the zoning objective and reproducible. The importance of this goal follows from the fact that morphostructural zoning, especially the scheme of lineaments, is the starting point of our approach to prediction of strong earthquakes; it is important also to location of some mineral deposits. The definition of large elements of relief is formalized in the first place. On the basis of their characteristics a territory is divided into three types of areas: blocks, lineaments and knots. A precise and objective location of knot and lineament positions is the final aim of our formalized scheme. The application of the suggested algorithm to Eastern Tien Shan is described in the conclusion of the paper, as an illustration.           ARK: https://n2t.net/ark:/88439/y085028 Permalink: https://geophysicsjournal.com/article/158 &nbsp